SIMILARITY OF WEIGHTED DIRECTED ACYCLIC GRAPHS

1. SIMILARITY OF WEIGHTED DIRECTED ACYCLIC GRAPHS Presented By: Jing Jin Supervisors: Dr. Virendra C. Bhavsar and Dr. Harold Boley Aug. 31, 2006

2. Outline • Introduction • Schema Matching • E-Business & Multi-agent System • Tree and Graph Similarity Techniques • The wDAG Similarity Algorithm • wDAG Representation • The Flow Chart • Main Functions • The Data Structure for Reusing Computation Results • Illustrative Example • Comparison of Typical Schema Matching Algorithms • Computational Results • Conclusion • Future Work • References

3. IntroductionSchema Matching Fig. 1 The match operation [30] • Schema matching is a fundamental problem in many application domains, such as e-Business, semantic web, data integration and semantic query processing.

4. IntroductionE-Business & Multi-agent System • The wide acceptance of advanced technologies from the semantic web and web services in e-Business. • A multi-agent system can provide a virtual marketplace for seller and buyer agents to conduct e-Business activities, where the match-making between them is a crucial step.

5. IntroductionTree and Graph Similarity Techniques • Much research has been done on tree [5, 21, 27, 32, 37, 39, 40] and graph [22, 23, 36, 41] similarity techniques. • The weighted tree similarity algorithm [5] Fig. 2 An example of two weighted trees describing second hand car information

6. IntroductionTree and Graph Similarity Techniques (cont’d) • The tree edit distance algorithm [21, 27, 39] Fig. 3 A tree edit distance example

7. Similarity flooding [23] IntroductionTree and Graph Similarity Techniques (cont’d) Fig. 4 The sequence of steps to determine the correspondences between tables and columns in S1 and S2

8. CUPID [22] IntroductionReusing Schema and Mapping Information Fig. 5 An example of lazy schema tree expansion

9. COMA [11, 31] IntroductionReusing Schema and Mapping Information (cont’d) Fig. 6 An example of reusing previous similarity value in COMA

10. Fig. 3 Weighted tree representation of a purchase

11. Fig. 4 The weighted tree from Fig. 3 folded into a wDAG representation

12. The wDAG Similarity Algorithm wDAG Representation Fig. 5 wDAG serialization in weighted OO RuleML

13. The wDAG Similarity AlgorithmThe Flow Chart Fig. 6 The flow chart of the wDAG similarity algorithm

14. The wDAG Similarity AlgorithmMain Functions Fig. 7 The comparison between wDAGs A and B

15. The wDAG Similarity AlgorithmMain Functions:wDAGsim(g, g’) The equation embedded in the main function wDAGsim(g, g’) that computes the similarity of two (sub)wDAGs is shown below.

16. The wDAG Similarity AlgorithmMain Functions:wDAGplicity(g) The equation embedded in the wDAGplicity(g) that computes the simplicity of the missing wDAG is shown below. where, DI: depth degradation index, 0 < DI  1.0 DF: depth degradation factor, 0 < DF  0.5 m: breadth of the wDAG g that is not a leaf. DI= 1.0 DF =0.5

17. The wDAG Similarity AlgorithmThe Data Structure for Reusing Computation Results • Two-dimensional Matrix • Adjacency Lists

18. The Data Structure for Reusing Computation ResultsTwo-dimensional Matrix • The matrix would be a sparse one which contains many NULL values. Table 1 The two-dimensional matrix corresponding to the wDAGs in Fig. 7 Table 3 Single-dimensional array of wDAG B Table 2 Single-dimensional array of wDAG A

19. The Data Structure for Reusing Computation ResultsAdjacency Lists Fig. 8 The structure of adjacency lists Fig. 9 The corresponding adjacency lists of the example in in Fig. 7

20. Illustrative Example

21. Table 4: Comparison of Typical Schema Matching Algorithms

22. Table 4: Comparison of Typical Schema Matching Algorithms (cont’d)

23. Table 4: Comparison of Typical Schema Matching Algorithms (cont’d)

24. Computational ResultsExample Set 1

25. Computational ResultsExample Set 1 (cont’d)

26. Example Set 1 (cont’d)

27. Example Set 1 (cont’d)

28. Example Set 2 Example 1 Example 2

29. Example Set 2 (cont’d) Example 2 Example 3

30. Example Set 2 (cont’d) Example 4 Example 5

31. Remarks • The size of OO RuleML source file of wDAG is smaller than that of weighted tree because wDAG eliminates the duplicate copies of shared substructures. And the time complexity of the wDAG algorithm is much lower than that of the weighted tree algorithm.

32. Conclusion • Mechanisms of schema matching approaches have been reviewed, and relevant tree and graph similarity techniques have been studied. • A wDAG similarity algorithm for the match-making in e-Business environments have been proposed. If two wDAGs are intuitively more similar, our similarity value is higher, otherwise, it is lower. • The intermediate similarity and simplicity values of shared sub-wDAGs can be reused efficiently through the adjacency lists structure. • The applicability and efficiency of the weighted tree similarity algorithm have been improved. • To our knowledge, in the field of match-making in e-Business environments, wDAG representations and their similarity measures have not been studied before.

33. Future Work • The test data is currently developed manually. A tool should be developed to provide a graphical user interface for users to define such data, like the Teclantic application of the AgentMatcher group (http://teclantic.ca/). • Improvements should be made on the simplicity function. One way is to replace the arithmetic mean by the geometric mean, as currently explored by the AgentMatcher group. • The proposed algorithm outputs the similarity between two wDAGs. The similarity values for multiple wDAGs should be ranked by some sorting algorithm, like bubble sort, just as the AgentMatcher group has already implemented ranking for tree similarity.

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36. Thank you! Mercy! 谢谢!